Is President Trump stable and competent enough to do his job?

"Republicans in Congress, the highest of intelligence officials, the highest of military officers in our country, leaders of the business community -- all of whom have dealt with the White House, and many of them dealt personally with Donald Trump -- have come to believe that he is unfit for the presidency," Bernstein told CNN's Brian Stelter on Sunday.

He said those people are "raising the very question of his stability and his mental fitness."

"Maybe what I'm being told is not as pervasive as I believe it is. Let's find out," Bernstein added. "We need as journalists to make this our primary function right now."

Stelter said on CNN's "Reliable Sources" that Trump's "actions and inactions in the wake of Charlottesville are provoking some uncomfortable conversations."

The questions, he said, include these: "Is the president of the United States a racist? Is he suffering from some kind of illness? Is he fit for office? And if he's unfit -- then what?"

Some Republican lawmakers, like U.S. Sen. Bob Corker, have joined Democrats in publicly raising questions about Trump's fitness for office. Last week the Tennessee Republican said that Trump "has not yet been able to demonstrate the stability nor some of the competence that he needs to demonstrate in order to be successful."

U.S. Rep.Zoe Lofgren, a California Democrat, recently said she wants Trump to undergo a mental health exam.

Another California Democrat, U.S. Rep.Jackie Speier, called for Trump's removal under the 25th Amendment.

On CNN's "State of the Union," Jake Tapper asked one of Speier's colleagues, U.S. Rep.Adam Schiff, about the tweet. He didn't go as far as she did, but he said "there are some serious issues with our president that aren't going to go away."

On "Reliable Sources," the conversation was about the appropriate way to address these concerns head-on.

Alice Stewart, a former journalist who is now a conservative commentator, said reporters should be discussing Trump's competency with their sources.

"I think if people are raising this question, it's a valid topic for journalists to cover," Stewart said. But she said what needs to be examined is "whether it is more of a political difference as opposed to a serious medical problem."

Stewart is the former communications director for U.S. Sen.Ted Cruz's presidential campaign. Cruz, a Texas Republican, sparred fiercely with Trump during the primary campaign last year.

She said Corker's recent remarks show that such questions aren't just coming from Trump's political opponents. They're coming from "across the aisle."